Well, that discomfort may seem religious, but segregationists felt justified by scripture too. They got over it; their churches got over it; so will yours.
It’s not that simple. The debate about race was very specific to America, modernity, the South. (Bans on interracial marriage were generally a white supremacist innovation, not an inheritance from Christendom or common law.) The slave owners and segregationists had scriptural arguments, certainly. But they were also up against one of the Bible’s major meta-narratives — from the Israelites in Egypt to Saint Paul’s “neither Jew nor Greek, slave nor free.”
That’s not the case with sex and marriage. The only clear biblical meta-narrative is about male and female. Sex is an area of Jewish law that Jesus explicitly makes stricter. What we now call the “traditional” view of sexuality was a then-radical idea separating the early church from Roman culture, and it’s remained basic in every branch of Christianity until very recently. Jettisoning it requires repudiating scripture, history and tradition in a way the end of Jim Crow did not.